Finite-size corrections to Poisson approximations in general renewal-success processes

Consider a renewal process, and let K⩾0 denote the random duration of a typical renewal cycle. Assume that on any renewal cycle, a rare event called “success” can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence can be relatively slow, because each success corresponds to a time interval, not a point. If K is an arithmetic variable, a “finite-size correction” (FSC) is known to speed Poisson convergence by providing a second, subdominant term in the appropriate asymptotic expansion. This paper generalizes the FSC from arithmetic K to general K. Genomics applications require this generalization, because they have already heuristically applied the FSC to p-values involving absolutely continuous distributions. The FSC also sharpens certain results in queuing theory, insurance risk, traffic flow, and reliability theory.